The sequential (distributional) topological complexity of the ordered configuration space of disks in a strip

TL;DR

This paper calculates the sequential topological complexity of the configuration space of disks in a strip, providing a formula for the minimum complexity of robot movement with intermediate stops.

Contribution

It introduces a formula for the r-th sequential topological complexity of the configuration space of disks in a strip, linking robot count, width, and movement complexity.

Findings

The complexity is r(n - ⌈n/w⌉) when n > w.

Any non-looping robot program with r-2 stops requires at least this many cases.

The results quantify the difficulty of planning robot paths in narrow aisles.

Abstract

How hard is it to program $n$ robots to move about a long narrow aisle while making a series of $r-2$ intermediate stops, provided only $w$ of the robots can fit across the width of the aisle? In this paper, we answer this question by calculating the $r^{\text{th}}$-sequential topological complexity of $\text{conf}(n,w)$, the ordered configuration space of $n$ open unit-diameter disks in the infinite strip of width $w$, as well as its $r^{\text{th}}$-sequential distributional topological complexity. We prove that as long as $n$ is greater than $w$, the $r^{\text{th}}$-sequential (distributional) topological complexity of $\text{conf}(n,w)$ is $r\big(n-\big\lceil\frac{n}{w}\big\rceil\big)$. This shows that any non-looping program moving the $n$ robots between arbitrary initial and final configurations, with $r-2$ intermediate stops, must consider at least $r\big(n-\big\lceil\frac{n}{w}\big\rceil\big)$ cases.